Count and Say Problem in DSA Typescript - Time & Space Complexity

Time Complexity: Count and Say Problem

O(2^n)

Understanding Time Complexity

We want to understand how the time needed to generate the "count and say" sequence grows as we ask for longer sequences.

Specifically, how does the work increase when we move from one term to the next?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function countAndSay(n: number): string {
  if (n === 1) return "1";
  const prev = countAndSay(n - 1);
  let result = "";
  let count = 1;
  for (let i = 1; i < prev.length; i++) {
    if (prev[i] === prev[i - 1]) {
      count++;
    } else {
      result += count.toString() + prev[i - 1];
      count = 1;
    }
  }
  result += count.toString() + prev[prev.length - 1];
  return result;
}
    

This code generates the nth term of the "count and say" sequence by recursively building the previous term and then reading it to form the current term.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The for-loop that reads the previous term to build the current term.
• How many times: The loop runs once for each character in the previous term, and the function calls itself recursively n times.

How Execution Grows With Input

Each term depends on the previous term's length, which grows roughly by a factor of about 1.3 each time.

Input Size (n)	Approx. Operations
5	~15 characters processed
10	~50 characters processed
15	~160 characters processed

Pattern observation: The length and work grow roughly exponentially, so the time grows faster than just doubling.

Final Time Complexity

Time Complexity: O(2^n)

This means the time roughly doubles with each increase in n, making it grow very fast as n gets bigger.

Common Mistake

[X] Wrong: "The time grows linearly because we just loop through the previous term once."

[OK] Correct: Each term depends on the previous term, which itself took time to build, so the total work adds up exponentially, not just linearly.

Interview Connect

Understanding how recursive calls and string building affect time helps you explain and optimize similar problems in interviews confidently.

Self-Check

"What if we changed the recursion to iteration, storing all previous terms? How would the time complexity change?"

Practice

1. What does the "Count and Say" sequence describe in simple terms?

easy

A. It describes each term by counting and saying the digits of the previous term.

B. It sorts the digits of the previous term in ascending order.

C. It adds the digits of the previous term to get the next term.

D. It reverses the digits of the previous term to form the next term.

Solution

1. Step 1: Understand the sequence definition

   The "Count and Say" sequence generates each term by describing the count of digits seen in the previous term.

2. Step 2: Compare options with definition

   Only It describes each term by counting and saying the digits of the previous term. correctly states this process; others describe unrelated operations.

3. Final Answer:

   It describes each term by counting and saying the digits of the previous term. -> Option A

4. Quick Check:

   Count and Say = Describe digits count [OK]

Hint: Remember: count digits, then say them aloud [OK]

Common Mistakes:

• Thinking it sorts digits instead of counting
• Assuming it sums digits for next term
• Confusing with reversing digits

2. Which of the following TypeScript function signatures correctly represents a function that takes a number n and returns the nth term of the Count and Say sequence as a string?

easy

A. function countAndSay(n: number): string

B. function countAndSay(n: string): number

C. function countAndSay(): number

D. function countAndSay(n: number): number[]

Solution

1. Step 1: Identify input and output types

   The function should take a number n (term index) and return a string representing that term.

2. Step 2: Match signature with requirements

   function countAndSay(n: number): string matches input as number and output as string; others mismatch types or parameters.

3. Final Answer:

   function countAndSay(n: number): string -> Option A

4. Quick Check:

   Input number, output string [OK]

Hint: Input is term number, output is sequence string [OK]

Common Mistakes:

• Using wrong input type (string instead of number)
• Returning number or array instead of string
• Missing input parameter

3. What is the output of the following TypeScript code snippet?

function countAndSay(n: number): string {
  if (n === 1) return "1";
  const prev = countAndSay(n - 1);
  let result = "";
  let count = 1;
  for (let i = 1; i <= prev.length; i++) {
    if (prev[i] === prev[i - 1]) {
      count++;
    } else {
      result += count + prev[i - 1];
      count = 1;
    }
  }
  return result;
}
console.log(countAndSay(4));

medium

A. "21"

B. "111221"

C. "312211"

D. "1211"

Solution

1. Step 1: Trace recursive calls and terms

   Term 1: "1"; Term 2: "11" (one 1); Term 3: "21" (two 1s); Term 4: "1211" (one 2, one 1).

2. Step 2: Confirm output for n=4

   The code returns "1211" for n=4, matching "1211".

3. Final Answer:

   "1211" -> Option D

4. Quick Check:

   Term 4 = "1211" [OK]

Hint: Remember sequence: 1, 11, 21, 1211... [OK]

Common Mistakes:

• Confusing term 3 and 4 outputs
• Off-by-one errors in loop
• Miscounting repeated digits

4. Identify the bug in this TypeScript implementation of the Count and Say sequence:

function countAndSay(n: number): string {
  if (n === 1) return "1";
  const prev = countAndSay(n - 1);
  let result = "";
  let count = 1;
  for (let i = 0; i < prev.length; i++) {
    if (prev[i] === prev[i + 1]) {
      count++;
    } else {
      result += count + prev[i];
      count = 1;
    }
  }
  return result;
}

medium

A. The count variable is not reset correctly inside the loop.

B. The loop should start from i = 1, not 0.

C. Accessing prev[i + 1] causes out-of-bounds error on last iteration.

D. The base case should return "0" instead of "1".

Solution

1. Step 1: Analyze loop boundary and indexing

   The loop runs from i=0 to prev.length-1, but accesses prev[i+1], which is undefined at last index.

2. Step 2: Identify error cause

   Accessing prev[i+1] at last iteration causes out-of-bounds access, leading to incorrect behavior or runtime error.

3. Final Answer:

   Accessing prev[i + 1] causes out-of-bounds error on last iteration. -> Option C

4. Quick Check:

   Check array bounds when accessing i+1 [OK]

Hint: Check array access at last index carefully [OK]

Common Mistakes:

• Ignoring last element boundary
• Starting loop at wrong index
• Incorrect base case value

5. Given the Count and Say sequence, what is the 6th term generated by the following TypeScript function?

function countAndSay(n: number): string {
  if (n === 1) return "1";
  const prev = countAndSay(n - 1);
  let result = "";
  let count = 1;
  for (let i = 1; i <= prev.length; i++) {
    if (prev[i] === prev[i - 1]) {
      count++;
    } else {
      result += count + prev[i - 1];
      count = 1;
    }
  }
  return result;
}
console.log(countAndSay(6));

hard

A. "13112221"

B. "312211"

C. "1113213211"

D. "21121314"

Solution

1. Step 1: List terms up to 6

   Term 1: "1"; Term 2: "11"; Term 3: "21"; Term 4: "1211"; Term 5: "111221"; Term 6: "312211".

2. Step 2: Confirm output matches term 6

   The function returns "312211" for n=6, matching "312211".

3. Final Answer:

   "312211" -> Option B

4. Quick Check:

   Term 6 = "312211" [OK]

Hint: Memorize first 6 terms or trace stepwise [OK]

Common Mistakes:

• Mixing up term 5 and 6
• Miscounting repeated digits
• Skipping recursive calls